Metric Spaces, Topological Spaces, and Compactness
... pick a convergent subsequence of (x 1ν ) in X1 and consider the corresponding subsequence of (xν ), which we relabel (xν ). Using this, pick a convergent subsequence of (x2ν ) in X2 . Continue. Having a subsequence such that x jν → yj in Xj for each j = 1, . . . , m, we then have a convergent subseq ...
... pick a convergent subsequence of (x 1ν ) in X1 and consider the corresponding subsequence of (xν ), which we relabel (xν ). Using this, pick a convergent subsequence of (x2ν ) in X2 . Continue. Having a subsequence such that x jν → yj in Xj for each j = 1, . . . , m, we then have a convergent subseq ...
MORE ON CONTRA ı-PRECONTINUOUS FUNCTIONS
... Since Cl.A/ ı Cl.A/ for any subset A of X , therefore, every contra-precontinuous is contra-ı-precontinuous but not conversely as following example shows. Example 4 ([5]). A contra-semi-continuous function need not be contra-precontinuous. Let f W R ! R be the function f .x/ D Œx, where Œx is th ...
... Since Cl.A/ ı Cl.A/ for any subset A of X , therefore, every contra-precontinuous is contra-ı-precontinuous but not conversely as following example shows. Example 4 ([5]). A contra-semi-continuous function need not be contra-precontinuous. Let f W R ! R be the function f .x/ D Œx, where Œx is th ...
Topology Proceedings - topo.auburn.edu
... functions f : X → R on a compact metric space X. Once it was observed that X could be represented as the maximal elements of the continuous dcpo UX , all other details were handled using domain theory. Naturally, if all we need to do to define an integral is represent a space in this manner, one won ...
... functions f : X → R on a compact metric space X. Once it was observed that X could be represented as the maximal elements of the continuous dcpo UX , all other details were handled using domain theory. Naturally, if all we need to do to define an integral is represent a space in this manner, one won ...
Lecture Notes on General Topology
... is Geometry (rubber sheet geometry). We call Set theory is the language of Topology. The course which we will study is basically known as Point Set Topology or General topology. To define Topology in an other way is the qualitative geometry. The basic idea is that if one geometric object can be cont ...
... is Geometry (rubber sheet geometry). We call Set theory is the language of Topology. The course which we will study is basically known as Point Set Topology or General topology. To define Topology in an other way is the qualitative geometry. The basic idea is that if one geometric object can be cont ...
Chapter 6 Convergences Preserving Continuity
... Hence, because of (‡) we obtain ∀t ∈ O, Fα (t) ⊆ c? (F(t)). Therefore ∀t ∈ O, F(t) ∩ U , ∅, because if for a t0 from O the set F(t0 ) ∩ U were an empty set, then Fα (t0 ) ⊆ c? (F(t0 )) = Y \ W̄ would lead to a contradiction. F is proved to be lower semicontinuous at a. ...
... Hence, because of (‡) we obtain ∀t ∈ O, Fα (t) ⊆ c? (F(t)). Therefore ∀t ∈ O, F(t) ∩ U , ∅, because if for a t0 from O the set F(t0 ) ∩ U were an empty set, then Fα (t0 ) ⊆ c? (F(t0 )) = Y \ W̄ would lead to a contradiction. F is proved to be lower semicontinuous at a. ...
a hit-and-miss hyperspace topology on the space of fuzzy sets
... The concept of a fuzzy set was introduced by L.A.Zadeh in [12] and later the concept of a fuzzy topology by C.L.Chang in [5]. After this introduction numerous authors have investigated how the properties of topological spaces can be extended to fuzzy topological spaces like compactness, connectednes ...
... The concept of a fuzzy set was introduced by L.A.Zadeh in [12] and later the concept of a fuzzy topology by C.L.Chang in [5]. After this introduction numerous authors have investigated how the properties of topological spaces can be extended to fuzzy topological spaces like compactness, connectednes ...
What Is...a Topos?, Volume 51, Number 9
... on the Ui ’s which coincide on the intersections Ui ∩ Uj . Now, let C be a category having finite projective limits. To give a topology (sometimes called a Grothendieck topology) on C means to specify, for each object U of C , families of maps (Ui → U)i∈I , called covering families, enjoying propert ...
... on the Ui ’s which coincide on the intersections Ui ∩ Uj . Now, let C be a category having finite projective limits. To give a topology (sometimes called a Grothendieck topology) on C means to specify, for each object U of C , families of maps (Ui → U)i∈I , called covering families, enjoying propert ...
The way-below relation of function spaces over semantic domains
... form a base for the Scott open sets. In this paper, continuous domains are always considered as topological spaces endowed with the Scott topology. With respect to this topology, a continuous domain L is sober and locally compact (in the sense that every point has a base of compact neighbourhoods), ...
... form a base for the Scott open sets. In this paper, continuous domains are always considered as topological spaces endowed with the Scott topology. With respect to this topology, a continuous domain L is sober and locally compact (in the sense that every point has a base of compact neighbourhoods), ...
On Semi-open Sets With Respect To an Ideal
... there is an open set U such that U ⊆ A ⊂ cl(U). This motivates our first definition. Definition 1. A subset A of X is said to be semi-open with respect to an ideal I (written as I -semi-open) if there exists an open set U such that U − A ∈ I and A − cl(U) ∈ I . If A ∈ I , then it is easy to see that ...
... there is an open set U such that U ⊆ A ⊂ cl(U). This motivates our first definition. Definition 1. A subset A of X is said to be semi-open with respect to an ideal I (written as I -semi-open) if there exists an open set U such that U − A ∈ I and A − cl(U) ∈ I . If A ∈ I , then it is easy to see that ...
IOSR Journal of Mathematics (IOSR-JM)
... The identity map f: (X, , I) (Y, ) is Irwg-continuous but not *- continuous. Theorem 3.5: Every continuous function is Irwg-continuous. Proof: Let f be a continuous function and V be a closed set in (Y, ) .Then f-1(V) is closed in (X, , I).Since every closed set is * -closed and hence Irwg – closed, ...
... The identity map f: (X, , I) (Y, ) is Irwg-continuous but not *- continuous. Theorem 3.5: Every continuous function is Irwg-continuous. Proof: Let f be a continuous function and V be a closed set in (Y, ) .Then f-1(V) is closed in (X, , I).Since every closed set is * -closed and hence Irwg – closed, ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.